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Numerical computation of the genus of an irreducible curve within an algebraic set. (English) Zbl 1211.14062

Summary: The common zero locus of a set of multivariate polynomials (with complex coefficients) determines an algebraic set. Any algebraic set can be decomposed into a union of irreducible components. Given a one-dimensional irreducible component, i.e. a curve, it is useful to understand its invariants. The most important invariants of a curve are the degree, the arithmetic genus and the geometric genus (where the geometric genus denotes the genus of a desingularization of the projective closure of the curve). This article presents a numerical algorithm to compute the geometric genus of any one-dimensional irreducible component of an algebraic set.

MSC:

14Q05 Computational aspects of algebraic curves
65H10 Numerical computation of solutions to systems of equations
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
14H99 Curves in algebraic geometry

Software:

Bertini; MultRoot
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Full Text: DOI

References:

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